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Unformatted text preview: MATH 2144 - FALL 2011 - EXAM I VERSION 1 NAME: SOLUTIONS The exam has 13 problems. The total number of points is 100. Be sure to follow the instructions for each problem. Draw a box around each of your final answers. Write neatly and legibly. Unreadable answers are wrong. 1. (8 points) After each function below state all the classes of functions to which that function belongs out of this list: power, root, polynomial, rational, algebraic, trigonometric, exponential, logarithmic . On each part of the problem you will receive one point for each correct answer and lose one point for each incorrect answer that you state (but not get less than zero). (a) x 5 + 2 polynomial, rational, algebraic (b) 1 3 x exponential (c) 1 + x 3 x 2 + 5 rational, algebraic (d) 1 + x 1 / 3 x 2 + 4 algebraic (e) q 1 + √ x algebraic 2. (4 points) State the domain of the function f ( x ) = √ 16- x 2 using interval notation. Solution : For the square root to be defined we need 16- x 2 ≥ 0. This is equivalent to x 2 ≤ 16. This is in turn equivalent to- 4 ≤ x ≤ 4. In interval notation this is written [- 4 , 4] . 3. (6 points) H ( x ) = q sin( x 4 ). Find f ( x ), g ( x ), and h ( x ) such that H = f ◦ g ◦ h . Do not use the identity function i ( x ) = x . Solution : Recall that ( f ◦ g ◦ h )( x ) = f ( g ( h ( x ))), so one first computes h ( x ), then feeds it into g to get g ( h ( x )), then feeds that into f to get f ( g ( h ( x ))). To identify f , g , and h think about how H ( x ) is built up from x . x → x 4 → sin( x 4 ) → q sin( x 4 ) Think of this as an assembly line in which something is produced from the input x by a sequence of three machines. The first machine h takes its input x and raises it to the fourth power so it can described as h ( x ) = x 4 . The second machine g takes the sine of its input x 4 to get sin( x 4 ). Note that the only thing it does with its input is to take its sine, so its job can be described by g ( x ) = sin x . The third machine f takes the square root of its input sin( x 4 ) to get q sin( x 4 ). Note that the only thing it does with its input is to take its square root, so its job can be described by f ( x ) = √ x . f ( x ) = √ x,g ( x ) = sin x , h ( x ) = x 4 4. (8 points) Find a formula for the inverse f- 1 ( x ) of the function f ( x ) = 3- 4 x 2 x- 1 . Show all the algebraic steps necessary to solve for f- 1 ( x ) and be sure to state the inverse as a function of x ....
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